When a straight line passes through point P (3,2), the inclination angle is twice that of the straight line x-4y + 3 = 0 A straight line passes through point P (3,2), intersects two points a and B with the positive half axis of X and Y axes, and the area of △ AOB is the smallest (o is the coordinate origin)

When a straight line passes through point P (3,2), the inclination angle is twice that of the straight line x-4y + 3 = 0 A straight line passes through point P (3,2), intersects two points a and B with the positive half axis of X and Y axes, and the area of △ AOB is the smallest (o is the coordinate origin)

Let the inclination angle of the line x-4y + 3 = 0 be a
Then the inclination angle of the straight line is 2A
tga=1/4
tg2a=2tga/(1-tga^2)=2*(1/4)/(1-1/16)=8/15
So the slope of the new line is 8 / 15
According to the point oblique formula: Y-2 = (x-3) * 8 / 15
8x-15y+6=0

If a and B satisfy a + 2B = 1, then the line ax + 3Y + B = 0 must pass the fixed point ()
A. （−16，12）B. （12，−16）C. （12，16）D. （16，−12）

Because a and B satisfy a + 2B = 1, then the line ax + 3Y + B = 0 becomes (1-2b) x + 3Y + B = 0, that is, x + 3Y + B (- 2x + 1) = 0 is constant, x + 3Y = 0 − 2x + 1 = 0, the solution is x = 12Y = − 16, so the line passes through the fixed point (12, − 16)

If the line y = K (x + 1) and the curve y = root (2x-x ^ 2) have a common point, then the value range of real number is --
Although it's not difficult, I think it's very difficult. The answer is wrong. Write down the calculation process,

Substituting y = K (x + 1) into the curve equation
So, K (x + 1) = √ (2x-x & sup2;)
The two sides are squared and simplified, and we get,
(k²+1)x²+2(k²-1)x+k²＝0
Δ＝[2(k²-1)]²-4(k²+1)k²＝-12k²+4
Let Δ ≥ 0, then, - 12K & sup2; + 4 ≥ 0, that is, 12K & sup2; - 4 ≤ 0
So, √ 3 / 3 ≤ K ≤ √ 3 / 3

Given that the slope of the line (2 + M-M ^ 2) x - (4-m ^ 2) y + m ^ 2-4 = 0 does not exist, then the value of M is________ .
The general formula with straight line is 4-m ^ 2 / 2 + M-M ^ 2 = - A / b
If the slope does not exist, shouldn't the numerator be equal to zero or the denominator be equal to zero
The solution should be positive or negative 2 or - 1, but the answer is - 2
Why?

If the slope does not exist, it means that the slope is infinite, and the line is perpendicular to the X axis
So the coefficient of X is not equal to 0, and the coefficient of Y is equal to 0
From 4-m ^ 2 = 0, M = 2 or - 2 can be obtained
When m = 2, the coefficient 2 + M-M ^ 2 of X is 0
So m can only take - 2